Generalized coordinate surrogates for integrated estimation and control

ABSTRACT

A robotic device includes a control system. The control system receives a first measurement indicative of a first distance between a center of mass of the machine and a first position in which a first leg of the machine last made initial contact with a surface. The control system also receives a second measurement indicative of a second distance between the center of mass of the machine and a second position in which the first leg of the machine was last raised from the surface. The control system further determines a third position in which to place a second leg of the machine based on the received first measurement and the received second measurement. Additionally, the control system provides instructions to move the second leg of the machine to the determined third position.

CROSS-REFERENCE TO RELATED APPLICATIONS

This U.S. patent application is a continuation of, and claims priorityunder 35 U.S.C. § 120 from, U.S. patent application Ser. No. 17/303,731,filed on Jun. 7, 2021 which is a continuation of, and claims priorityunder 35 U.S.C. § 120 from, U.S. patent application Ser. No. 16/055,983,filed on Aug. 6, 2018, which is a divisional of, and claims priorityunder 35 U.S.C. § 121 from, U.S. patent application Ser. No. 14/834,855,filed on Aug. 25, 2015, which claims priority under 35 U.S.C. § 119(e)to U.S. Provisional Application 62/041,275, filed on Aug. 25, 2014. Thedisclosures of these prior applications are considered part of thedisclosure of this application and are hereby incorporated by referencein their entireties.

BACKGROUND

As technology advances, various types of robotic devices are beingcreated for performing a variety of functions that may assist users.Robotic devices may be used for applications involving materialhandling, transportation, welding, assembly, and dispensing, amongothers. Over time, the manner in which these robotic systems operate isbecoming more intelligent, efficient, and intuitive. As robotic systemsbecome increasingly prevalent in numerous aspects of modern life, it isdesirable for robotic systems to be efficient. Therefore, a demand forefficient robotic systems has helped open up a field of innovation inactuators, movement, sensing techniques, as well as component design andassembly.

SUMMARY

The present application discloses implementations that relate tocontrolling a position of a leg of a robotic device to balance therobotic device. A robotic device may include a control device thatemploys a control procedure referred to as “touch-down lift-off” (TDLO).The TDLO control procedure may involve determining a location at whichto place a particular leg of the robotic device based on measurementsassociated with another leg. The control device may move the particularleg of the robotic device to the determined location in order tomaintain balance and/or the velocity of the robotic device.

An example implementation may include receiving, by a machine, a firstmeasurement indicative of a first distance between a center of mass ofthe machine and a first position in which a first leg of the machinelast made initial contact with a surface. The implementation may alsoinclude receiving, by the machine, a second measurement indicative of asecond distance between the center of mass of the machine and a secondposition in which the first leg of the machine was last raised from thesurface. The implementation may further include determining a thirdposition in which to place a second leg of the machine based on thereceived first measurement and the received second measurement.Additionally, the implementation may include providing instructions tomove the second leg of the machine to the determined third position.

In another example, the present application describes a non-transitorycomputer-readable medium having stored thereon instructions that, uponexecution by at least one processor, performs a set of operations. Theoperations may include receiving, by a machine, a first value indicativeof a first distance between a center of mass of the machine and a firstposition in which a first leg of the machine last made contact with asurface. The operations may also include receiving, by the machine, asecond value indicative of a second distance between the center of massof the machine and a second position in which the first leg of themachine was last raised from the surface. The operations may furtherinclude determining a third position in which to place a second leg ofthe machine based on the received first value and the received secondvalue. Additionally, the operations may include providing instructionsto move the second leg of the machine to the determined third position.

In yet another example, the present application describes a machine thatincludes at least two legs, a sensor, and a control device configured toperform a set of operations. The at least two legs includes a first legand a second leg. The operations may include receiving, from the sensor,a first measurement indicative of a first distance between a center ofmass of a machine and a first position in which the first leg of themachine last made contact with a surface. The operations may alsoinclude receiving, from the sensor, a second measurement indicative of asecond distance between the center of mass of the machine and a secondposition in which the first leg of the machine was last raised from thesurface. The operations may further include determining a third positionin which to place the second leg of machine based on the received firstmeasurement and the received second measurement. Additionally, theoperations may include providing instructions to move the second leg ofthe machine to the determined third position.

In one aspect, the present application describes a system. The systemmay include a means for receiving, by a machine, a first measurementindicative of a first distance between a center of mass of the machineand a first position in which a first leg of the machine last madecontact with a surface. The system may also include a means forreceiving, by the machine, a second measurement indicative of a seconddistance between the center of mass of the machine and a second positionin which the first leg of the machine was last raised from the surface.The system may further include a means for determining a third positionin which to place a second leg of the machine based on the receivedfirst measurement and the received second measurement. Additionally, thesystem may include a means for providing instructions to move the secondleg of the machine to the determined third position.

These as well as other aspects, advantages, and alternatives, willbecome apparent to those of ordinary skill in the art by reading thefollowing detailed description, with reference where appropriate to theaccompanying figures.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates a configuration of a robotic system, according to anexample implementation.

FIG. 2 illustrates a perspective view a quadruped robot, according to anexample implementation.

FIG. 3 illustrates a side view of a robotic leg, according to an exampleimplementation.

FIG. 4A illustrates a side view of states of robotic legs of a roboticdevice, according to an example implementation.

FIG. 4B illustrates a schematic top-down view of states of robotic legsof a robotic device, according to an example implementation.

FIG. 5 is a flowchart of an example TDLO procedure, according to anexample implementation.

FIG. 6A, FIG. 6B, and FIG. 6C are graphics illustrating foot positionsrelative to a robotic device's center of mass over time, according to anexample implementation.

FIG. 7A illustrates a top-down view of straight-line TDLO stepping,according to an example implementation.

FIG. 7B illustrates a top-down view of TDLO stepping with a constantyaw, according to an example implementation.

FIG. 8 is a computer-readable medium configured according to an exampleimplementation.

DETAILED DESCRIPTION

The following detailed description describes various features andfunctions of the disclosed systems and methods with reference to theaccompanying figures. The illustrative system and method embodimentsdescribed herein are not meant to be limiting. It may be readilyunderstood that certain aspects of the disclosed systems and methods canbe arranged and combined in a wide variety of different configurations,all of which are contemplated herein.

An example implementation involves a robotic device configured with atleast two robotic legs, sensors, and a control device. The controldevice may implement a control system that instructs the movement of therobotic legs to balance the robotic device. The control device mayreceive measurements or data from the sensors indicative of theposition, velocity, acceleration, and/or orientation of each robotic legat a given time. Each robotic leg may have any number of actuators(e.g., 2-5 actuators) that can be controlled to move the robotic legthrough its range of motion.

The control device may implement a control procedure that is referred toherein as “touch-down lift-off” (TDLO). TDLO involves determining aposition in which a stance leg of a robotic device first made initialcontact with the ground (touch-down) and a position in which the sameleg was raised from the ground (lift-off). The position measurements arerelative to the center of mass (CoM) of the robot. For example, when arobot's body is moving above a stationary planted leg, the touch-downand lift-off measurements differ although the leg is not being dragged,lifted, or otherwise moved from its planted position. TDLO also involvesdetermining a position at which to place a swing leg of the roboticdevice in order to maintain balance. A control device implementing TDLOmay control actuators on the swing leg to move it to the determinedposition. In some implementations, the control device may provideinstructions to a computing device that controls the movement of therobotic leg.

During operation, at a given time, a legged robotic device may have oneor more “stance” legs and one or more “swing” legs. For example, awalking bipedal robot generally has one leg planted on the ground (thestance leg) as the other leg (the swing leg) is lifted from its previousplanted position and moved to the next planted position on the ground.In a typical walking gait for a bipedal robot, each leg alternatesbetween being a stance leg and being a swing leg.

As the bipedal robot walks forward, a stance leg is first planted infront of the robot. Then, the robot's body moves over the planted stanceleg until the stance leg is behind the robot's body. As this occurs, theother leg (the swing leg) swings forward and is planted in front of therobot's body. Collectively, the period of time where one leg is a stanceleg and the other is a swing leg is referred to herein as a “stanceperiod” (consecutive stance periods denoted herein as k, (k+1), etc.).When the swing leg is planted in front of the robot, it becomes thestance leg for the next stance period, and the previous stance leg islifted from its planted position and begins swinging forward to becomethe swing leg for the following stance period. In other words, if theleft leg was the stance leg and the right leg was the swing leg instance period k, the left leg will be the swing leg and the right legmay be the stance leg in stance period (k+1).

TDLO considers two measurements for a stance leg—the touch-downmeasurement (the position of the stance leg relative to the robot's CoMat the moment the stance leg makes initial contact with the ground) andthe lift-off measurement (the position of the stance leg relative to therobot's CoM at the moment the stance leg lifts off the ground). Based onthese two measurements for the stance leg, the TDLO control lawdetermines a position at which to place the swing leg (which, onceplaced on the ground, becomes the next stance leg). Then, TDLO involvesmeasuring the touch-down and lift-off positions for the new stance legto determine the position at which to place the new swing leg.

In some implementations, a robotic leg may alternatively or additionallymodify the magnitude and location of forces being applied to the groundby the robotic device to maintain balance. In some cases, moving the legmay not be necessary to maintain balance, and the control system mayalternatively apply a greater or lesser extent of force to the ground tomaintain balance.

In some implementations, the TDLO control procedure may include aforcing function that modifies the behavior of a control deviceimplementing the TDLO control procedure. A robotic device may includeany number of robotic legs that are naturally offset from the CoM of therobotic device (i.e., not directly beneath the CoM of the roboticdevice). These offsets may specify a forcing function, therebypreventing the control device from providing extraneous correction tobalance the robotic device. In some instances, the forcing function mayalter a robot's stepping pattern to achieve a more natural movement ofthe robotic legs.

In some cases, the forcing function may bias the direction in which therobotic device is travelling. For instance, a certain trajectory may bedesired, and the forcing function may be used to guide the roboticdevice along that trajectory. As one possible example, a robotic devicemay be instructed to move in a particular direction with a specifiedspeed, and the forcing function may account for this desired speed anddirection in order to move the robot this way while maintaining balance.

The control system may be configured to actuate one or more actuatorsconnected across components of a robotic leg. The actuators may becontrolled to raise or lower the robotic leg. In some cases, a roboticleg may include actuators to control the robotic leg's motion in threedimensions.

Referring now to the figures, FIG. 1 illustrates an exampleconfiguration of a robotic device, according to an exampleimplementation. The robotic system 100 represents an example roboticdevice configured to perform the operations described herein.Additionally, the robotic device 100 may be configured to operateautonomously, semi-autonomously, and/or using directions provided byuser(s), and may exist in various forms, such as a humanoid robot or aquadruped robot, among other examples. Furthermore, the robotic device100 may also be referred to as a robotic device, mobile robot, or robot,among other designations.

As shown in FIG. 1 , the robotic device 100 includes processor(s) 102,data storage 104, program instructions 106, controller 108, sensor(s)110, power source(s) 112, mechanical components 114, electricalcomponents 116, and electrical components 118. Note that the roboticdevice 100 is shown for illustration purposes, as robotic device 100 andmay include more or less components within examples. The variouscomponents of robotic device 100 may be connected in any manner,including wired or wireless connections, etc. Further, in some examples,components of the robotic device 100 may be positioned on multipledistinct physical entities rather on a single physical entity. Otherexample illustrations of robotic device 100 may exist as well.

Processor(s) 102 may operate as one or more general-purpose processorsor special purpose processors (e.g., digital signal processors,application specific integrated circuits, etc.). The processor(s) 102can be configured to execute computer-readable program instructions 106that are stored in the data storage 104 and are executable to providethe operations of the robotic device 100 described herein. For instance,the program instructions 106 may be executable to provide operations ofcontroller 108, where the controller 108 may be configured to causeactivation and deactivation of the mechanical components 114 and theelectrical components 116. The processor(s) 102 may operate and enablethe robotic device 100 to perform various functions, including thefunctions described herein.

The data storage 104 may exist as various types of storage configured tohold memory. For example, the data storage 104 may include or take theform of one or more computer-readable storage media that can be read oraccessed by processor(s) 102. The one or more computer-readable storagemedia can include volatile and/or non-volatile storage components, suchas optical, magnetic, organic or other memory or disc storage, which canbe integrated in whole or in part with processor(s) 102. In someembodiments, the data storage 104 can be implemented using a singlephysical device (e.g., one optical, magnetic, organic or other memory ordisc storage unit), while in other embodiments, the data storage 104 canbe implemented using two or more physical devices, which may communicatevia wired or wireless communication. Further, in addition to thecomputer-readable program instructions 106, the data storage 104 mayinclude additional data such as diagnostic data, among otherpossibilities.

The robotic device 100 includes at least one controller 108, which mayinterface with the robotic device 100. The controller 108 may serve as alink between portions of the robotic device 100, such as a link betweenmechanical components 114 and/or electrical components 116. In someinstances, the controller 108 may serve as an interface between therobotic device 100 and another computing device. Further, the controller108 may serve as an interface between the robotic system 100 and auser(s). The controller 108 may include various components forcommunicating with the robotic device 100, including a joystick(s),buttons, among others. The controller 108 may perform other functionsfor the robotic device 100 as well. Other examples of controllers mayexist as well.

Additionally, the robotic device 100 includes one or more sensor(s) 110such as force sensors, proximity sensors, motion sensors, load sensors,position sensors, touch sensors, depth sensors, ultrasonic rangesensors, and infrared sensors, among other possibilities. The sensor(s)110 may provide sensor data to the processor(s) 102 to allow forappropriate interaction of the robotic system 100 with the environmentas well as monitoring of operation of the systems of the robotic device100. The sensor data may be used in evaluation of various factors foractivation and deactivation of mechanical components 114 and electricalcomponents 116 by controller 108 and/or a computing system of therobotic device 100.

The sensor(s) 110 may provide information indicative of the environmentof the robotic device for the controller 108 and/or computing system touse to determine operations for the robotic device 100. For example, thesensor(s) 110 may capture data corresponding to the terrain of theenvironment or location of nearby objects, which may assist withenvironment recognition and navigation, etc. In one exampleconfiguration, the robotic device 100 may include a sensor system thatmay include a camera, RADAR, LIDAR, a global positioning system (GPS)transceiver, and/or other sensors for capturing information of theenvironment of the robotic device 100. The sensor(s) 110 may monitor theenvironment in real-time and detect obstacles, elements of the terrain,weather conditions, temperature, and/or other parameters of theenvironment for the robotic device 100.

Further, the robotic device 100 may include other sensor(s) 110configured to receive information indicative of the state of the roboticdevice 100, including sensor(s) 110 that may monitor the state of thevarious components of the robotic device 100. The sensor(s) 110 maymeasure activity of systems of the robotic device 100 and receiveinformation based on the operation of the various features of therobotic device 100, such the operation of extendable legs, arms, orother mechanical and/or electrical features of the robotic device 100.The sensor data provided by the sensors may enable the computing systemof the robotic device 100 to determine errors in operation as well asmonitor overall functioning of components of the robotic device 100. Forexample, the computing system may use sensor data to determine thestability of the robotic device 100 during operations, as well asmeasurements related to power levels, communication activities,components that require repair, among other information. As one exampleconfiguration, the robotic device 100 may include gyroscope(s),accelerometer(s), and/or other possible sensors to provide sensor datarelating to the state of operation of the robotic device. Further,sensor(s) 110 may also monitor the current state of a function, such asa gait, that the robotic system 100 may currently be operating.Additionally, the sensor(s) 110 may measure a distance between a givenrobotic leg of a robotic device and a center of mass of the roboticdevice. Other example uses for the sensor(s) 110 may exist as well.

Additionally, the robotic device 100 may also include one or more powersource(s) 112 configured to supply power to various components of therobotic device 100. Among possible power systems, the robotic device 100may include a hydraulic system, electrical system, batteries, and/orother types of power systems. As an example illustration, the roboticdevice 100 may include one or more batteries configured to provide powerto components that may be charged via a wired and/or wirelessconnection. Within examples, components of the mechanical components 114and electrical components 116 may each connect to a different powersource or may be powered by the same power source. Components of therobotic system 100 may connect to multiple power sources as well.

Within example configurations, any type of power source may be used topower the robotic device 100, such as a gasoline engine. Further, thepower source(s) 112 may charge using various types of charging, such aswired connections to an outside power source, wireless charging,combustion, or other examples. Other configurations may also bepossible. Additionally, the robotic device 100 may include a hydraulicsystem configured to provide power to the mechanical components 114using fluid power. Components of the robotic device 100 may operatebased on hydraulic fluid being transmitted throughout the hydraulicsystem to various hydraulic motors and hydraulic cylinders, for example.The hydraulic system of the robotic device 100 may transfer a largeamount of power through small tubes, flexible hoses, or other linksbetween components of the robotic device 100. Other power sources may beincluded within the robotic device 100.

Mechanical components 114 represent hardware of the robotic system 100that may enable the robotic device 100 to operate and perform physicaloperations. As a few examples, the robotic device 100 may includeactuator(s), extendable leg(s) (“legs”), arm(s), wheel(s), one ormultiple structured bodies for housing the computing system or othercomponents, and/or other mechanical components. The mechanicalcomponents 114 may depend on the design of the robotic device 100 andmay also be based on the operations and/or tasks the robotic device 100may be configured to perform. As such, depending on the operationsand/or tasks of the robotic device 100, different mechanical components114 may be available for the robotic device 100 to utilize. In someexamples, the robotic device 100 may be configured to add and/or removemechanical components 114, which may involve assistance from a userand/or other robotic device. For example, the robotic device 100 may beinitially configured with four legs, but may altered by a user or therobotic device 100 to remove two of the four legs to operate as a biped.Other examples of mechanical components 114 may be included within someexamples.

The electrical components 116 may include various components capable ofprocessing, transferring, providing electrical charge or electricsignals, for example. Among possible examples, the electrical components116 may include electrical wires, circuitry, and/or wirelesscommunication transmitters and receivers to enable operations of therobotic device 100. The electrical components 116 may interwork with themechanical components 114 to enable the robotic device 100 to performvarious operations. The electrical components 116 may be configured toprovide power from the power source(s) 112 to the various mechanicalcomponents 114, for example. Further, the robotic device 100 may includeelectric motors. Other examples of electrical components 116 may existas well.

In some implementations, the robotic device 100 may also includecommunication link(s) 118 configured to send and/or receive information.The communication link(s) 118 may transmit data indicating the state ofthe various components of the robotic device 100. For example,information read in by sensor(s) 110 may be transmitted via thecommunication link(s) 118 to a separate device. Other diagnosticinformation indicating the integrity or health of the power source(s)112, mechanical components 114, electrical components 116, processor(s)102, data storage 104, and/or controller 108 may be transmitted via thecommunication link(s) 118 to an external communication device. Otherinformation may be transmitted over the communication link(s) 118 aswell.

In some implementations, the robotic device 100 may receive informationat the communication link(s) 118 that is processed by the processor(s)102. The received information may indicate data that is accessible bythe processor(s) 102 during execution of the program instructions 106,for example. Further, the received information may change aspects of thecontroller 108 that may affect the behavior of the mechanical components114 or the electrical components 116. In some cases, the receivedinformation indicates a query requesting a particular piece ofinformation (e.g., the operational state of one or more of thecomponents of the robotic device 100), and the processor(s) 102 maysubsequently transmit that particular piece of information back out thecommunication link(s) 118.

In some cases, the communication link(s) 118 include a wired connection.The robotic device 100 may include one or more ports to interface thecommunication link(s) 118 to an external device. The communicationlink(s) 118 may include, in addition to or alternatively to the wiredconnection, a wireless connection. Some example wireless connections mayutilize a cellular connection, such as CDMA, EVDO, GSM/GPRS, or 4Gtelecommunication, such as WiMAX or LTE. Alternatively or in addition,the wireless connection may utilize a Wi-Fi connection to transmit datato a wireless local area network (WLAN). In some implementations, thewireless connection may also communicate over an infrared link,Bluetooth, or a near-field communication (NFC) device.

FIG. 2 illustrates a perspective view of an example quadruped robot,according to an example embodiment. Among other possible functions, therobotic device 200 may be configured to perform the procedures describedherein during operation. The robotic device 200 includes legs 206A,206B, 206C, and 206D connected to a body 202 of the robotic device 200and may also include sensors configured to provide sensor data to acomputing system of the robotic device 200. Sensors may be incorporatedwithin portions of the legs 206A, 206B, 206C, and 206D to detectpressure, position, velocity, acceleration, and orientation, among othermeasurements. Sensors may be in other locations as well. Further, therobotic device 200 is illustrated carrying objects 208 on the body 202.Within other example implementations, the robotic device 200 may includemore or fewer components and may additionally include components notshown in FIG. 2 .

The robotic device 200 may be a physical representation of the roboticsystem 100 shown in FIG. 1 or may be based on other configurations. Tooperate, the robotic device 200 includes a computing system that be madeup of one or multiple computing devices configured to assist in variousoperations of the robotic device 200. These operations may includeprocessing data and providing outputs based on the data to cause variouschanges to the physical state of the robotic device 200. The computingsystem may process information provided by various systems of therobotic device 200 (e.g., a sensor system) or from other sources (e.g.,a user, another robotic device, a server) and may provide instructionsto these systems to operate in response.

Additionally, the computing system may monitor systems of the roboticdevice 200 during operation, which may include monitoring for errorsand/or monitoring regular operation, for example. In some exampleconfigurations, the computing system may serve as a connection betweenthe various systems of the robotic device 200, and may coordinate theoperations of the systems together to enable the robotic device 200 toperform tasks. Further, the computing system may include multipledevices, processors, controllers, and/or other entities configured tocontrol or assist in the operation of the robotic device. Additionally,the computing system may operate using various types of memory and/orother components.

The robotic device 200 exists as a quadruped robotic device with fourextendable legs 206A, 206B, 206C, and 206D. Although the robotic device200 includes four legs 206A, 206B, 206C, and 206D in the illustrationshown in FIG. 2 , the robotic device 200 may include more or less legswithin other examples. Further, the configuration, position, and/orstructure of the legs 206A, 206B, 206C, and 206D may vary in exampleimplementations. The legs 206A, 206B, 206C, and 206D enable the roboticdevice 200 to move and may be configured to operate in multiple degreesof freedom to enable different techniques of travel to be performed. Inparticular, the legs 206A, 206B, 206C, and 206D may enable the roboticdevice 200 to travel at various speeds by way of mechanical control.

A gait is a pattern of movement of the limbs of animal, robotic device,or other mechanical structure. As such, the robotic device 200 maynavigate by operating the legs 206A, 206B, 206C, and 206D to performvarious gaits. The robotic device 200 may use a variety gaits to travelwithin an environment, which may involve selecting a gait based onspeed, terrain, the need to maneuver, and/or energy efficiency. Therobotic device 200 may be configured to dynamically switch betweengaits, which may enable the robotic device to change speeds or themechanics of operating the legs 206A, 206B, 206C, and 206D.

Further, different types of robotic devices may use different gaits dueto differences in design that may enable or prevent use of certaingaits. Although some gaits may have specific names (e.g., walk, trot,run, gallop, and bound), the distinctions between these gaits may beslight. The gaits may be classified based on footfall patterns, thepatterns including locations on a surface for the placement of distalends of the extendable legs (e.g., feet). Similarly, gaits may also beclassified based on mechanics.

Additionally, the robotic device 200 may include other mechanicalapertures or appendages, which may be attached to the robotic device 200at various positions. The robotic device 200 may include mechanicalarms, grippers, wheels, or other features. The legs 206A, 206B, 206C,and 206D may have feet or other types of mechanical features (e.g.,rubber feet to increase friction between the robotic device and asurface) that enables control upon various types of surfaces that therobotic device may encounter.

As part of the design of the example robotic device 200, the body 202 ofthe robotic device 200 connects to the legs 206A, 206B, 206C, and 206Dand may house various components of the robotic device 200. As such, thestructure of the body 202 may vary within examples and may furtherdepend on particular operations that a given robotic device may havebeen designed to perform. For example, a robotic device developed tocarry heavy loads may have a wide body that enables placement of theloads. Similarly, a robotic device designed to reach high speeds mayhave a narrow, small body that does not have substantial weight.Further, the body 202 as well as the legs 206A, 206B, 206C, and 206D maybe developed using various types of materials, such as various metals orplastics. Within other examples, a robotic device may have a body with adifferent structure or made of other types of materials.

The sensor(s) 204 of the robotic device 200 may include various types ofsensors, such as a camera or a sensing system. The sensor(s) 204 may beplaced at various positions on the robotic device. As described for therobotic system 100, the robotic device 200 may include a sensory systemthat includes a camera, RADAR, LIDAR, a GPS transceiver,accelerometer(s), gyroscope(s), and/or other types of sensors. Thesensor(s) may be configured to measure parameters of the environment ofthe robotic device 200 as well as monitor internal operations of systemsof the robotic device 200. In some examples, the robotic device 200 mayinclude sensors to measure the orientation, position, velocity, oracceleration of each leg 206A, 206B, 206C, and 206D.

The loads 208 carried by the robotic device 200 may represent varioustypes of cargo that the robotic device 200 may transport. The loads 208may also represent external batteries or other types of power sources(e.g., solar panels) that the robotic device 200 may utilize. The loads208 represent one example use the robotic device 200 may be configuredfor. As such, the robotic device 200 may be configured to perform otheroperations as well.

Additionally, as shown with the robotic system 100, the robotic device200 may also include various electrical components that may enableoperation and communication between the mechanical features of therobotic device 200. As previously indicated, the robotic device 200 mayinclude one or more computing systems that include one or multipleprocessors configured to perform various functions, including processinginputs to provide outputs. The computing system may include additionalcomponents, such as various types of storage and a power source, etc.

Control system 202 of robotic device 200 may cause the robotic device200 to navigate an environment based on sensor data from the sensingsystem. The sensing system may include sensors of sensing system coupledto portions of the robotic device 200. The robotic device 200 mayreceive navigation commands by way of a communication system. Forinstance, the robotic device may receive a command to move forward at 5kilometers per hour. The command may specify to walk forward for aparticular distance, such as 100 meters. In one example, a command mayspecify one or more locations at which to place particular legs.

In some examples, the navigation commands may involve GPS coordinates.In one instance, a command may instruct the robotic device to navigateto a particular position, which may be defined by particular GPScoordinates. The robotic device may then cause the locomotion system tomove to the position while navigating physical features of the terrainidentified by the control system (perhaps based on data from perceptionsensors). Another command may instruct the robotic device to follow aparticular person, who may have with them a GPS enabled device thatgenerates data indicating the position of the person. The data may becommunicated to the robotic device that may then cause the locomotionsystem to follow the person while navigating physical features of theterrain identified by the control system.

In some example embodiments, during operation, the computing system maycommunicate with other systems of the robotic device 200 via wired orwireless connections and may further be configured to communicate withone or multiple users of the robotic device. As one possibleillustration, the computing system may receive an input from a userindicating that the user wants the robotic device to perform aparticular gait in a given direction. The computing system may processthe input, and may cause the systems of the robotic device to performthe requested gait. Additionally, the robotic device's electricalcomponents may include other type of electrical components, includingbut not limited to interface, wires, busses, and/or other communicationlinks configured to enable systems of the robotic device to communicate.

Furthermore, the robotic device 200 may communicate with one or moreusers and/or other robotic devices via various types of interfaces. Inan example embodiment, the robotic device 200 may receive input from auser via a joystick or similar type of interface. The computing systemmay be configured to receive data indicative of the amount of force andother possible information from a joystick interface. Similarly, therobotic device 200 may receive inputs and communicate with a user viaother types of interface, such as a mobile device or a microphone.Regardless, the computing system of the robotic device 200 may beconfigured to process the various types of inputs that the roboticdevice 200 may receive.

FIG. 3 is a side-view of an example articulable robotic leg 300. Therobotic leg 300 includes a member 302 having an inboard end that isconnected to the robotic device at joint 308. The member 302 has anoutboard end that is connected in a rotatable manner to an inboard endof a member 304 at joint 306. The member 304 has an outboard end that isconnected to a foot member 314. The robotic leg 300 also includes anactuator 312 connected between the member 302 and the robotic device. Insome implementations, the actuator 312 may be a linear hydraulicactuator. Actuation of the actuator 312 causes the member 302 and themember 304 to rotate around joint 308. Similarly, actuation of theactuator 310 causes the member 304 to rotate around the joint 306.

Actuating the actuator 310 and the actuator 312 in combination may causethe leg to take a step. For instance, the actuator 310 may retract,which causes member 304 to rotate counter-clockwise (from theperspective shown in FIG. 3 ) around joint 306. This rotation may raisethe leg 300 up from the ground. The actuator 312 may retract, whichcauses member 302 to rotate clockwise (from the perspective shown inFIG. 3 ) around joint 308. By rotating member 302 clockwise around joint308, foot member 314 moves forward relative to the ground. Actuators 310and 312 may then extend and thereby cause robotic leg 300 to lower andpush against the ground, thereby causing the robotic device to moveforward or to adopt a new stance. Example states of the robotic leg 300are depicted in FIG. 4A and FIG. 4B.

The robotic device may move according to different gaits by varying thetiming of actuation, speed of actuation, and range of actuation of theactuators. The particular gaits that a particular robotic device iscapable of performing may depend upon the range of motion of its legsand the force and velocity specifications of the actuators. The range ofmotion of its legs may in turn depend upon the leg length and range oftravel of the actuators. In some implementations, the robotic deviceincludes hydraulic actuators, which may provide linear actuation by wayof pressurized fluid within a chamber and a piston. Acceleration of suchactuators is proportional to the pressure of the hydraulic fluid used toactuate the hydraulic actuator—with a given load, higher pressureresults in greater acceleration. The control system may select aparticular gait based on factors such as speed, terrain, the need tomaneuver, and/or energy efficiency. For instance, the robotic device maytransition from a walk to a run as speed of locomotion is increased. Therobotic device may then transition back to a walk on uneven terrain. Therobotic device may include any combination of hydraulic actuators,pneumatic actuators, electric motors, thermal actuators, magneticactuators, and/or other mechanical actuators.

Load on the hydraulic actuators may vary during the stepping sequence.During the portion of the gait in which the hydraulic actuators arecausing a leg to push against the ground, the load on the hydraulicactuators is relatively large compared to the portion of the gait inwhich the hydraulic actuators are raising the leg and stepping forward.As the load varies, the robotic device may vary the hydraulic pressureto maintain the movement of the legs according to the gait.

Although the side view of the robotic leg in FIG. 3 is shown with twoactuators 310 and 312 that move the robotic leg in two dimensions, therobotic leg may have any number of actuators that allow for more orfewer degrees of freedom. In some cases, the robotic leg may includeactuators that allow for lateral movement of the robotic leg (i.e., inthe y-direction depicted in FIG. 3 ) in addition to longitudinalmovement (i.e., in the x-direction depicted in FIG. 3 ) and verticalmovement (i.e., in the z-direction depicted in FIG. 3 ). Further, therobotic leg is depicted as having two members 302 and 304, however anynumber of members may be included that might be connected at joints andcontrolled with actuators.

FIG. 4A and FIG. 4B depict example states of robotic legs of a roboticdevice. The robotic legs shown in FIG. 4A may be the same as or similarto robotic leg 300 depicted in FIG. 3 . At a given point in time, therobotic leg may be lifting off a surface (which may be referred toherein as the “lift-off state”), swinging forward, touching down on asurface (which may be referred to herein as the “touch-down state”), orplanted on the ground, among other possible states. In operation, therobotic leg may transition between these various states by extending orcompressing the actuators of the robotic leg. The moment of lift-off andthe moment of touch-down may occur at different points in time. In FIG.4A, the stance leg during stance period k is illustrated using solidlines, and the swing leg during stance period k is illustrated usingdotted lines. A robotic device may have any number of robotic legs thatmay be moved in a similar way to the robotic leg depicted in FIG. 4A.The robotic device may be in motion or stationary, depending on theparticular scenario.

In some implementations, a robotic device may include a control devicethat receives information from sensors on various portions of therobotic device and controls the actuators of a particular leg in orderto move that leg to a particular location. The control device may usesensor information to determine the location at which to place a givenrobotic leg.

The robotic legs depicted in FIGS. 3 and 4A are shown from a side-viewperspective and depict single-dimensional measurements and movement ofthe robotic device (in the x-direction). However, in some scenarios, arobotic device may have a trajectory that involves lateral turning ortraversing sloped surfaces. The robotic leg may move within athree-dimensional space, and the measurements and determinations of theplacement of the foot of a robotic leg may be performed for eachdimension to provide balance control in a variety of scenarios.

The robotic leg depicted in FIG. 4A may be a simplified representationof a robotic leg, and that, in various implementations, a robotic legmay be differently proportioned or may contain additional components.Additionally, the body of the robotic device is a simplifiedrepresentation of a body for the purposes of explanation. Further, thesurface depicted in FIG. 4A may be referred to as “the ground,” althoughthe surface that robotic device stands on may be any other man-made ornatural surfaces.

FIG. 4A illustrates two robotic legs of a robotic device 400 at twopoints in time-first state 410 and second state 420. The robotic device400 shown in FIG. 4A is moving in a forward x-direction (i.e., to theright in FIG. 4A). At first state 410, the body of the robotic device400 is in position 412 and has a CoM 414. The first state 410 mayrepresent a particular point in time at which the swing leg is inposition 416 and the stance leg is in position 418. At the point in timerepresented by first state 410, the stance leg in position 418 has madeinitial contact with the ground, at a distance of x_(TD,k) from the CoM414 of the robotic device 400. Additionally, the point in timerepresented by first state 410 is the moment at which the swing leg inposition 416 is about to be lifted off the ground and begin movingforward. The first state 410 marks the beginning of stance period k.

After the stance leg in position 418 is placed on the ground and theswing leg in position 416 is raised from the ground, the roboticdevice's body pivots on the stance leg and moves from position 412 toposition 422. The weight of robotic device 400 is placed upon the stanceleg as it transitions from the position 418 in the first state 410 toposition 428 in the second state 420. The point in time represented bysecond state 420 is the moment at which the stance leg in position 428is about to be lifted off the ground and begin moving forward, therebybecoming the next swing leg. The distance x_(LO,k) from the CoM 424 ofthe robotic 400 represents the lift-off measurement during stance periodk. While the stance leg is planted, the swing leg lifts from position416 and swings forward until it reaches position 426. A control deviceof the robotic device 400 implementing TDLO control law may use thex_(TD,k) and the x_(LO,k) measurements as a basis for determining theparticular position 426, which corresponds to a distance x_(TD,k)+1 infront of the CoM 424 of the robotic device 400 at state 420. At themoment in time where the stance leg in position 428 is lifted from theground and the swing leg in position 426 is planted on the ground, thestance period advances to (k+1).

Note that the gait of the robotic device 400 described above is an idealwalking gait, where exactly one stance leg is touching the ground at anygiven point in time, and the swing leg is in the air swinging to thenext touchdown. At stance transitions, as a swing leg makes initialcontact with the ground, the stance leg is lifted from the ground at thesame moment in time. However, in some scenarios, a slower walking gaitmay result both the swing leg and the stance leg touching the groundsimultaneously for a short period of time before the swing leg is liftedand begins moving forward. In other scenarios, a faster gait (e.g.,running) may result in neither the swing leg nor the stance leg touchingthe ground at certain points in time during a given stance period. TDLOcontrol law may also be applied to these various gaits; the idealwalking gait description above is provided for explanatory purposes.

In some instances, the location of the foot relative to the ground maychange between the first state 410 and the second state 420. Forexample, the surface may be slick or otherwise provide a low amount offriction, causing the foot to slip and change its location on theground. Thus, the location at which the foot of the robotic leg ispivoted on the ground relative to the ground may differ between firststate 410 and second state 420. In some implementations, sensors (e.g.,an IMU on the robotic device's body) may detect these slips and providecorrective measures to the TDLO control law in order to maintainstability.

In some instances, it may be desired to move the robotic device at aconstant velocity. In these instances, states 410 and 420 may beseparated by a fixed time interval (e.g., 0.1 seconds to 5 seconds). Onthe other hand, it may be desired to accelerate and/or decelerate therobotic device in order to increase or decrease its velocity. In somecases, the time interval between states may be shortened or lengthenedin order to increase or decrease the robotic device's velocity,respectively. In other cases, the time interval between states may beheld at the fixed time interval, and the distances between touch-downx_(TD,k) and lift-off x_(LO,k) may be decreased or increased in order toincrease or decrease the robotic device's velocity, respectively. Insome implementations, both the time interval separating states and thedistances x_(TD,k) and x_(LO,k) may be modified in order to accelerateor decelerate the robotic device.

Note that, although the touch-down position x_(TD,k) can be determinedat the moment the robotic leg makes initial contact with the groundduring stance period k, the final lift-off position x_(LO,k) for stanceperiod k may not be known until the moment the robotic leg lifts off theground. Thus, the final lift-off measurement x_(LO,k) for the stanceleg-which is used to determine the next touch-down location x_(TD,k+1)of the swing leg in stance period k (i.e., the next stance leg in stanceperiod (k+1))—may not be determined until a moment before the swing legtouches down.

In some instances, a control system of the robotic device mayperiodically measure the lift-off location x_(LO,k) during a givenstance period. When the lift-off location x_(LO,k) is updated, a newtouch-down location x_(TD,k+1) may be calculated or determined. Thisprocess of periodically measuring the lift-off location andre-calculating the next touch-down location may be performed one or moretimes during a given stance period. In some implementations, the controlsystem may operate actuators of the swing leg when the touch-downlocation is updated in order to adjust for the updated touch-downlocation.

In scenarios where a robotic device has a walking gait where at leastone leg is on the ground at a given point in time, the next touch-downlocation may be updated periodically during a given stance period untilthe swing leg makes initial contact with the ground at or near the mostrecently determined new touch-down location (based in part on the mostrecently measured lift-off location). In scenarios where a roboticdevice has a running gait where both legs may be off the ground atcertain points in time, the control system may use the final lift-offmeasurement (taken at the moment the stance leg broke contact with theground) in order to determine the next touch-down measurement.

FIG. 4B illustrates a schematic top-down view 430 of a robotic devicemoving in a forward x-direction (upward with respect to FIG. 4B).Aspects of the illustration in FIG. 4B may correspond to the roboticdevice 400 shown in FIG. 4A, provided from a different perspective. Inthe top-down view 430, the robotic device has two legs-Leg B along theleft side of FIG. 4B, and Leg A along the right side of FIG. 4B. Therobotic device's CoM is shown at various points in time and isillustrated using a circle with an “X.” Three stance periods—k, (k+1),and (k+2)—are shown in FIG. 4B. The foot placement of each stance leg(i.e., the touch-down location) is illustrated using a concentric circlesurrounding a filled circle along the line of the stance leg.

During stance period k, Leg A is the stance leg and Leg B is the swingleg. Leg A is placed at a distance x_(TD,k) in front of the CoM 414 ofthe robotic device. The robotic device then moves forward over theplanted Leg A, swinging Leg B forward until the point in time at whichthe robotic device has a CoM 424. At that moment, Leg A is lifted fromits planted position at a distance x_(LO,k) from the CoM 424. A controldevice of the robotic device then determines the position at which toplace Leg B-x_(TD,k+1)—from the CoM 424 using TDLO control law, as shownin FIG. 4B. The robotic device then places Leg B at the determinedlocation in front of CoM 424, and the stance period advances to (k+1).Note that the touchdown and lift-off positions may be positive ornegative values. In this example, the touchdown positions are positivex-values (relative to the body center of mass at the moment oftouchdown), while the lift-off positions are negative x-values (relativeto the body center of mass at the moment of lift-off).

During stance period (k+1), Leg A becomes the swing leg and Leg Bbecomes the stance leg. The robotic device moves forward while Leg B isplanted on the ground until its CoM reaches position 434. At thatmoment, Leg B is lifted from its planted position at a distancex_(LO,k+1) from the CoM 434. The control device of the robotic devicethen determines the position at which to place Leg A—x_(TD,k+2)—usingTDLO control law, as illustrated in FIG. 4B. The robotic device alsoplaces Leg A at the determined location in front of CoM 434, and thestance period advances to (k+2).

Note that, while FIGS. 4A and 4B show two legs-a stance leg and a swingleg-some robotic devices implementing TDLO control law may have multipleswing legs and/or stance legs at a given time. Some implementations ofTDLO may consider two legs of a multi-legged robot a “pair” because theyare complementary (based on their position with respect to each other onthe robotic device, for example). For the purposes of this application,TDLO control law as described herein relates to bipedal robotic devicesand/or multi-legged robotic devices whose legs can be complementarilypaired.

In FIGS. 4A and 4B, the robotic device is moving in a one-dimensionaldirection along a flat surface, and thus the touch-down and lift-offpositions are shown to be x-directional positions. However, the roboticdevice may be travelling in two dimensions, and the touch-down andlift-off positions may have x-components and y-components. Thus, thecontrol device may determine the next touch-down position x_(TD,k+1) fortwo dimensions.

The TDLO control law and techniques used to modify the behavior of acontrol system implementing TDLO are discussed in further detail withrespect to FIG. 5 .

FIG. 5 is a flowchart of an example TDLO procedure, according to anexample implementation. The operations in the flowchart could, forexample, be performed by the robotic device 100 in FIG. 1 , the roboticdevice 200 in FIG. 2 , or any combination of components from roboticdevice 100 and robotic device 200. FIG. 5 may include one or moreoperations or procedures as illustrated by one or more of blocks 502,504, 506, and 508. Although the blocks are illustrated in a sequentialorder, these blocks may in some instances be performed in parallel,and/or in a different order than those described herein. Also, thevarious blocks may be combined into fewer blocks, divided intoadditional blocks, and/or removed based upon the desired implementation.

In addition, for FIG. 5 and other procedures and operations describedherein, the flowchart shows the functionality and operation of onepossible implementation. In this regard, each block may represent amodule, a segment, or a portion of program code, which includes one ormore instructions executable by a processor for implementing specificlogical operations or steps in this process. The program code may bestored on any type of computer-readable medium, for example, such as astorage device including a disk or hard drive. The computer-readablemedium may include a non-transitory computer-readable medium, forexample, such as computer-readable media that stores data for shortperiods of time like register memory, processor cache and random accessmemory (RAM). The computer-readable medium may also include othernon-transitory media, such as secondary or persistent long term storage,like read only memory (ROM), optical or magnetic disks, compact-discread only memory (CD-ROM), for example. The computer-readable media mayalso be any other volatile or non-volatile storage system. Thecomputer-readable medium may be considered a computer-readable storagemedium, a tangible storage device, or other article of manufacture, forexample. The program code (or data for the code) may also be stored orprovided on other media including communication media. For instance, thecommands may be received on a wireless communication media.

Further, for FIG. 5 and other procedures and operations describedherein, each block may represent circuitry that is arranged to performthe specific logical operations.

Operations of FIG. 5 may be fully performed by a control device, or maybe distributed across multiple control devices. In some examples, thecontrol device may receive information from sensors of a robotic device,or the control device may receive the information from a processor thatcollects the information. The control system might further communicatewith a remote control system (e.g., a control device on another roboticdevice) to receive information from sensors of other devices, forexample. For the purposes of explanation, the operations of blocks 502,504, 506, and 508 are described as being performed by a control device;however, other devices as described above may carry out the operations.

The following operations of blocks 502, 504, 506, and 508 describe usingmeasurements recorded within a particular stance period in order todetermine foot placement in the following stance period. As describedabove, this may involve taking measurements from a stance leg and usingthem as a basis to determine the placement of a swing leg. Upon movingthe swing leg to the determined position and planting its foot on theground, that swing leg becomes the stance leg for the following stanceperiod.

At block 502, a control device receives a first measurement indicativeof a first distance between a CoM of a machine and a first position inwhich a first leg of the machine made initial contact with the surface.The first leg may refer to the stance leg in the previous stance period.In some cases, the position in which the stance leg of the machineinitiated contact with the surface may be similar to the position 418 ofthe stance leg depicted in FIG. 4A. The measurement may be similar to orbased on x_(TD,k) depicted in FIG. 4A. As described herein, “initialcontact” may refer to the point in time at which the foot of a roboticleg transitions from swinging forward to being planted on the ground.Note that each step may have an “initial contact.”

The control device may be configured to receive the first measurementfrom one or more sensors coupled to portions of the robotic device. Thesensors may measure the position, velocity, acceleration, andorientation of the legs of the robotic device. In an exampleimplementation, each robotic leg has sensors coupled to the joints ofthat leg that measure the angles of those joints. The control device mayhave stored thereon a model of the robotic device that correlates a setof joint angles with a distance that the stance leg is from the CoM ofthe robotic device. Using known dimensions of the components of a givenrobotic leg and its connection point to the robotic device's body, acontrol device can determine the location of the foot of the robotic legwith respect to the robotic device's body and/or CoM. This technique ofdetermining foot position based on joint angles may be referred to as“forward kinematics.” The control device may then provide joint anglesmeasured at a time in which the stance leg initiates contact with asurface to the model to determine the first measurement. In anotherexample implementation, each robotic leg of the robotic device includesan inertial measurement unit (IMU) that senses the robotic leg'svelocity, orientation, and acceleration. Regardless of the particularimplementation, the position of the foot of the robotic leg with respectto the robotic device's body and/or CoM may be determined from sensorinformation and, taken at the moment at which the robotic leg makesinitial contact with the ground, serve as the first measurement.

In some cases, the foot positions may be measured with respect togravity. Put differently, the direction of the gravitational force(e.g., due to Earth's gravity) may define an axis or set of axes againstwhich the touchdown and lift-off measurements may be recorded. Forexample, the direction of the force of gravity may point in the negativez-direction, and a plane normal to that gravity vector may define therobotic device's x-y plane. In some implementations, the direction ofgravity may be measured or otherwise determined using a sensor, such asan IMU. Thus, if a robotic device is walking along a sloped surface(e.g., a surface with some degree of inclination or changes in height),the touchdown and lift-off locations may be measured with respect to aplane normal to the gravity vector (rather than a plane defined bysloped or inclined surface).

In some implementations, a robotic device is configured to include asensing system. The sensing system may receive data from a plurality ofsensors and calculate a position, orientation, and velocity of eachrobotic leg and/or feet of that robotic leg. The sensing system may alsodetermine the position, orientation, and velocity of the CoM of therobotic device. The sensing system may then provide the calculated oneor more of the position, orientation, and velocity measurements to thecontrol device. The sensing system may use any combination ofmeasurements from, for example, an IMU coupled to the robotic device'sbody and/or various joint angle sensors in order to determine theposition, orientation, velocity, and/or acceleration of one or moreparts of the robotic device. A control device of the robotic device mayconsider the robotic device's body orientation, velocity, and/oracceleration in determining the next touch-down location that improvesthe robotic device's balance. For instance, if the robotic device's bodyis pitched forward (i.e., tilted downward in the forward direction), thetouch-down location of the next stance leg may need to be modified totake into account the body orientation.

At block 504, a control device receives a second measurement indicativeof a second distance between the CoM of a machine and a second positionin which the first leg (e.g., the stance leg) of the machine leaves thesurface. In some cases, the position in which the stance leg of themachine last broke contact with the surface may be similar to theposition 428 of the stance leg in FIG. 4A. The measurement may besimilar to x_(LO,k) depicted in FIG. 4A. As described herein, “lastraised” may refer to the point in time at which the foot of a roboticleg transitions from being planted on the ground to lifting off theground and swinging forward.

Similarly to the operation at block 502, the control device may beconfigured to receive the second measurement from one or more sensorscoupled to portions of the robotic device. As described above, thesensors may measure the position, velocity, acceleration, andorientation of the legs of the robotic device. In some cases, thecontrol device receives these measurements and calculates or otherwisedetermines the distance between the CoM of the robotic device and theposition in which the stance leg of the machine was raised from thesurface. In some implementations, the control device may receive thesecond measurement from a sensing system, such as the previouslydescribed sensing system.

At block 506, the control device determines a third position in which toplace a second leg of the machine based on the received firstmeasurement and the received second measurement. The second leg mayrefer to the swing leg in the previous stance period. The control devicemay implement an open-loop or closed-loop controller that utilizes thefirst measurement and the second measurement as feedback to determinethe following touch-down position. The control device may determine thetouch-down position for the second leg in order to achieve balancecontrol. Further, the control device may provide estimations of eachrobotic leg's velocity and acceleration to other devices or systemscoupled to the robotic device. In some cases, the third position inwhich to place the second leg of the machine may be similar to position426 of the swing leg in FIG. 4A.

In order to achieve robust balance sensing and control, a “touch-downlift-off” (which may be referred to herein as “TDLO”) control law may beemployed by a control circuit, a computing device, or a programmablelogic device (PLD) such as a field-programmable gate array (FPGA). TheTDLO control equation is based on a linear-inverted pendulum (LIP) modeladapted to legged robotic devices that utilizes two time-separatedmeasurements of a position of a robot's stance leg relative to the CoMof the robotic device in order to determine foot placement of therobot's swing leg to achieve balance control and/or other control goals.The results of the TDLO control equation also provide information aboutthe state of the robot's balance. Additionally, the TDLO controlequation outputs may be used as a basis to control the leg of the robot.

The equation governing TDLO control law may be used to determine aposition in which to next place a swing leg in order to maintain thebalance of the robot. A legged robot-such as a bipedal robot or aquadruped robot, among others—may be configured to have legs that arenot directly beneath the robot's CoM. In robotic devices with four ormore legs, the TDLO control equation may measure the touch-down andlift-off locations of two or more stance legs and use those measurementsas a basis to determine the following touch-down positions for two ormore swing legs. Thus, in some implementations, the TDLO controlequation may be modified to use a forcing function or other offsetfactor to embed the leg configuration of the robotic device to produce amore natural stepping pattern. The forcing function may also be adjustedto influence the trajectory of a legged robot, which may cause therobotic device to laterally turn or move forward. Further, the forcingfunction may be modified to alter the robot's pace and gait.

In some implementations, the TDLO control equation calculates a nextfoot location for a swing leg based on linear combinations of twoprevious leg position measurements from a stance leg and a pair of gaincoefficients. The first measurement indicates a position in which thestance leg made initial contact with the ground, which is denoted hereinas x_(TD,k), where TD represents “touch down” and k represents adiscrete stance period associated with the most recent measurement. Thesecond measurement indicates a position in which the same leg was raisedfrom the ground, which is denoted herein as x_(LO,k), where LOrepresents “lift off” The values of the gain coefficients, which aredenoted as j_(x) and b_(x), may be selected or otherwise calculated inorder to stabilize the robot and to provide a desired effective naturalfrequency and amount of damping. In some embodiments, the gaincoefficients j_(x) and b_(x) are determined using the height of arobot's CoM above the ground and a nominal stance duration k asparameters of LIP control. Selection of gain coefficients j_(x) andb_(x) is discussed below in greater detail.

The TDLO control equation may be written as:

x _(TD,k+1) =j _(x)(x _(TD,k) −x _(LO,k))−b _(x)(x _(TD,k) +x_(LO,k))  [1]

The TDLO control equation [1] determines the next position at which toplace the next stance leg in the x-dimension. The TDLO equation abovemay be applied to other dimensions, such as the y-dimension, todetermine the position at which to place the leg in the y- as well. Thex- and y-, components of the position at which to place the leg may becombined, along with the z-height of the ground, to represent athree-dimensional position, and the control device may subsequentlycontrol the robotic leg to move to the determined three-dimensionalposition. In various implementations, the TDLO control law may onlyconsider measurements taken in two dimensions, such as the x- andy-dimensions, while not explicitly recording and controlling the legplacement on the basis of z-dimensional measurements.

Additionally, portions of the TDLO control equation [1] areapproximately proportional to the velocity v_(x) and acceleration a_(x)of the robot's center-of-mass. These may be written as:

v _(x) ˜x _(TD,k) −x _(LO,k)  [2]

a _(x)˜−(x _(TD,k) −x _(LO,k))  [3]

In this manner, TDLO can be thought of as being similar toproportional-derivative (a.k.a. “PD”) control on robot center-of-massvelocity.

The velocity and acceleration of the center-of-mass implied by portionsof the TDLO control equation [1] of the robotic device may be used toinform other control or estimation systems coupled to the robot. Thevelocity and acceleration estimations provided by the TDLO equations donot require differentiation of the measured position of a leg of therobotic device and thus is less susceptible to noise. Further, unlikeexplicit velocity calculations that differentiate position measurements,the estimated velocity can be quickly calculated, and, with the LIPmodel, removes much of the phase lag typically resulting from explicitvelocity calculations, leading to better stability of the roboticdevice.

In equation [1] above, x_(TD,k+1) represents the location in which toplace the next stance leg in order to maintain balance. Thus, at block506, the control device can use the first measurement x_(TD,k) and thesecond measurement x_(LO,k) as a basis for determining the next locationto place the next stance leg of the robotic device x_(TD,k+1). The gainvalues j_(x) and b_(x) are related to the desired dynamic response ofthe system to a disturbance, which could be represented by an effectivenatural frequency and damping ratio, and may be adjusted to increase ordecrease the rate and shape of disturbance decay. In an exampleoperation, if a robotic device experiences an externally applied lateralforce, the natural frequency and damping ratio determines how quickly,and with what oscillation, the robotic device will return to itsoriginal position and/or velocity. The damping ratio may be adimensionless number between 0 (representing an undamped system) and anypositive number, where a damping ratio between 0 and 1 represents anunderdamped system, a damping ratio equal to 1 represents a criticallydamped system, and a damping ratio greater than 1 represents anoverdamped system. Disturbance responses with higher desired naturalfrequencies will correct more quickly for tracking errors. Because TDLOoperates on the discrete stepping system of a legged robot, acritically-damped response, with sufficiently high natural frequency,will approach a “dead-beat” response in which the system will return tothe desired set-point (i.e., at equilibrium) in the minimum possibletime. Example plots depicting TDLO responses for an underdamped system,an overdamped system, and a dead-beat system are illustrated in FIG. 6A,FIG. 6B, and FIG. 6C, respectively.

In some embodiments, the gain values j_(x) and b_(x), are calculated toprovide stable balance control for a LIP. The approximate height of therobotic device's CoM, the stance duration (i.e., the length of time foreach discrete k), the desired natural frequency of the velocityresponse, and a desired damping ratio may all be used in calculatingstable gain values j_(x) and b_(x).

The following section describes one method of choosing the controlgains, j_(x) and b_(x), using pole-placement (i.e., eigenvalues) of thefinal discrete closed-loop system. In calculating these gaincoefficients, the legged robot is considered to behave similarly to aLIP. In the following equations, g represents acceleration due togravity and is equal to 9.81 m/s², z is equal to the height of therobotic device's CoM (in meters), T_(s) is equal to the stance durationper discrete k (in seconds), fn is equal to the desired naturalfrequency of the velocity response (in Hz), λ_(n) is the same frequency(in radians per second), and dr represents the desired damping ratio.Additionally, λ represents the modeled pendulum “tipping” frequency (inradians per second) of the LIP.

$\begin{matrix}{\lambda = \sqrt{\frac{g}{z}}} & \lbrack 4\rbrack\end{matrix}$ $\begin{matrix}{\lambda_{n} = {2\pi^{*}fn}} & \lbrack 5\rbrack\end{matrix}$

Equations [6] and [7] are used to determine continuous-time poles s₁ ands₂:

s ₁ =−dr*λ _(n)+λ_(n)√{square root over (dr ²−1)}  [6]

s ₂ =−dr*λ _(n)−λ_(n)√{square root over (dr ²−1)}  [7]

Note that poles s₁ and s₂ may be real numbers when the specified dampingratio dr is greater than or equal to 1, and complex numbers when thedamping ratio dr is less than 1. Next, another set of intermediatevalues p₁ and p₂, which represent poles of the discrete step-to-stepsystem in the transfer function used to calculate gain values j_(x) andb_(x), are calculated using equations [8] and [9], respectively.

p₁=e^(T) _(s) ^(*) _(s) _(1 [) 8]

p₂=e^(T) _(s) ^(*) _(s) _(2 [) 9]

Using these pole-placement values, determined from the desired fn anddr, and the other determined parameters, the gain values j_(x) and b_(x)may be calculated using equations [8] and [9]. Such calculations mayprovide gain values that stabilize a LIP.

$\begin{matrix}{j_{x} = \frac{\left( {1 + e^{2\lambda T_{s}} + {2e^{\lambda T_{s}}*\left( {{- p_{1}} - p_{2} + {p_{1}p_{2}}} \right)}} \right)}{2\left( {{- 1} + e^{\lambda T_{s}}} \right)^{2}}} & \lbrack 8\rbrack\end{matrix}$ $\begin{matrix}{b_{x} = \frac{\left( {1 + e^{2\lambda T_{s}} - {2e^{\lambda T_{s}}*\left( {p_{1} + p_{2} + {p_{1}p_{2}}} \right)}} \right)}{2\left( {1 + e^{\lambda T_{s}}} \right)^{2}}} & \lbrack 9\rbrack\end{matrix}$

At block 508, the control device provides instructions to move the nextstance leg of the machine to the determined third position.

In some implementations, the instructions are provided directly to theactuators on the robotic device to control the placement of the leg ofthe robotic device. As noted above, the instructions may be computerprogram instructions to a computing device that controls the movement ofthe robotic leg. In some implementations, the instructions may beprovided directly from the control device to actuators on the roboticleg. In some instances, an actuator is controlled by supplying an analogor digital signal to a dedicated driver that operates the actuator. Inother instances, the control device incorporates the circuitry ordigital logic used to operate the actuator, such that the control devicemay extend or retract each actuator without a separate driver.

The instructions to move the leg of the robotic device may include anynumber of digital or analog signals that operate electromechanicaldevices on the robotic leg, such as a hydraulic, pneumatic, orelectrical actuator. The signals may be provided to various componentsin a synchronized manner to lift the robotic leg and place it at thedetermined touch-down location that maintains the balance of the roboticdevice.

FIGS. 6A, 6B, and 6C illustrate example plots of TDLO responsesconfigured to have various damping ratios. The plots are oriented suchthat the horizontal axis indicates the foot position (e.g., in meters)with respect to a robotic device's CoM. A continuous line or curve thatsweeps horizontally is indicative of the CoM of the robotic device“swaying” or moving above the planted foot. The vertical axis indicatesa progression through time, where the top of the plot indicates thediscrete stance interval k=0, and moving downward through the plotadvances k. Each filled-in dot at one end of a curve represents a pointof touch down, while each empty dot at the other end of the curverepresents a point of lift off. A dot that is far from the center axisof x=0 indicates that the foot position is far from the CoM of therobotic device. Conversely, a dot that is near or aligned with thecenter axis of x=0 indicates a foot position that is nearly or directlybelow the robotic device's CoM. Thus, a stable TDLO response convergesto the center axis of x=0.

For the purposes of explanation, x indicates a left-right position of afoot with respect to the robotic device's CoM. The x values representdisplacement in one-dimension. Any number of dimensions (e.g., twodimension or three dimensions) may be factored into a particular TDLOresponse. The single-dimension examples disclosed herein are providedfor explanatory purposes, and are not meant to limit TDLO control to onedimension.

FIG. 6A illustrates an example plot 600 of an underdamped TDLO responseusing a damping ratio of 0.5. The plot 600 depicts the robotic device'sCoM “swaying” above the planted foot; that is, the CoM is moving in onedirection until the following step, at which point the CoM begins movingin the opposite direction. The foot position with respect to the roboticdevice's CoM (which may be simply referred to herein as “relative footposition”) in response sways from being on one side of the roboticdevice's CoM to the other side of the robotic device's CoM in order tocorrect the perturbation. The foot placement determined by theunderdamped TDLO control attenuates this swaying from stance intervalk=0 to k=6. In this example, it is not until k=7 that the steady-staterelative foot position beneath the robotic device's CoM is achieved.

FIG. 6B illustrates an example plot 610 of an overdamped TDLO responseusing a damping ratio of 2. The plot 610 depicts the robotic device'sCoM moving in one direction (in this example, the positive x-direction)above the planted foot. After k=0, the relative foot position inresponse moves in the same direction as the robotic device's motion toslow the motion of the robotic device's CoM. The foot placementdetermined by the overdamped TDLO control reduces the robotic device'sCoM velocity by progressively taking shorter steps in the positivex-direction until the robotic device slows to a halt from stanceinterval k=0 to k=6. In this example, like the underdamping TDLOcontrol, it is not until k=7 that the steady-state relative footposition beneath the stationary robotic device's CoM is achieved. Notethat, in this overdamped TDLO control example, the robotic device'sending stationary position may be some distance from the roboticdevice's initial position.

FIG. 6C illustrates an example plot 620 of a dead-beat TDLO responseusing a damping ratio of 1. The plot 620 depicts the robotic device'sCoM initially moving in the positive x-direction during k=0, thenslowing down (e.g., accelerating in the negative x-direction) duringk=1, and finally coming to rest at k=2. Unlike the underdamped andoverdamped TDLO control examples shown in FIGS. 6A and 6B, the dead-beatTDLO response stabilizes the robotic device in a shorter time (i.e., infewer stance durations k). While such a dead-beat response may appear tobe ideal, there may be physical limitations on a robotic device's jointsand actuators that would make achieving this rapid of a stabilizationdifficult. Note that some robotic devices may be capable of achieving adead-beat or near dead-beat response, depending upon the physicalattributes of the robotic device's joints and actuators. Furthermore,underdamped TDLO or overdamped control may be desired, depending on thearrangement of a particular robotic device.

Note that the time axis may not represent a continuous time axis. Whilethe plots shown in FIGS. 6A, 6B, and 6C show the touch-down and lift-offpoints occurring simultaneously, a response from a physicallyimplemented TDLO-controlled robotic device might have a separation intime between touch-down and lift-off. Both the time axis and the footposition axis may not necessarily be drawn to scale or correspond to aparticular set of gain values, damping ratio, natural frequency ofvelocity response, or stance height. The plots shown in FIGS. 6A, 6B,and 6C are provided to facilitate understanding and not necessarily torepresent any particular set of values or specific scenario.

The above-described TDLO principles and techniques may be extended toinclude velocity control and/or yaw control. One TDLO control techniqueinvolves stepping in order to bring the robotic device to a stop. Thismay be accomplished by regulating the robotic device's touchdownpositions to converge to a location below the robot's CoM. In thisexample, the robotic device's “heading-frame”—that is, a virtualreference frame that serves as a basis for TDLO-based calculations—maybe defined by the robot's CoM. The swing leg's next touchdown positionmay therefore be determined on the basis of two metrics: (i) the stanceleg's most recent touchdown position with respect to the robot's CoM atthe moment the touchdown occurred, and (ii) the stance leg's most recentliftoff position with respect to the robot's CoM at the moment theliftoff occurred. Thus, the two metrics may represent measurements takenin two different heading-frames corresponding to two different points intime.

This heading-frame paradigm may serve as an extension to TDLO forcontrolling the robotic device's speed and direction. Consider anexample scenario where a robotic device is instructed to walk at aconstant speed. A constant, straight-line (e.g., no turning or changesin yaw) walk may be accomplished by designating the heading-frame to alocation near the robotic device's CoM. For example, instead ofmeasuring the touchdown and liftoff stance foot positions with respectto the robotic device's CoM, the touchdown and liftoff positions may bemeasured with respect to a location “behind” the robotic device's CoM.Whereas TDLO using a CoM-centered heading-frame may cause the robot toslow to a stop, TDLO using a CoM-offset heading-frame may cause therobot to maintain a particular velocity (which may be proportional tothe extent of offset from the robot's CoM). In this manner, the TDLOcontrol techniques described herein may be extended to cause the roboticdevice to maintain a constant speed.

The heading-frame paradigm may also extend TDLO to allow for yawcontrol. Consider an example scenario where a robotic device isinstructed to walk in a circle (e.g., a constant speed and a constantyaw rate). Such a walk may be accomplished by utilizing a heading-framethat is both offset from the robotic device's CoM (to provide theconstant forward walking speed) and is configured to rotate at aparticular yaw rate. Thus, the heading-frame at the moment of the mostrecent stance leg liftoff may be rotated compared to the heading-frameat the moment of the most recent stance leg touchdown (because theheading-frame is configured to rotate over time, and the twomeasurements are separated in time). When the heading-frame is rotatedover time, TDLO may respond by attempting to compensate for this virtualrotation by stepping in a manner that causes the robot to turn.

FIGS. 7A and 7B illustrate top-down views of example TDLO-based walkingscenarios. The dotted rectangles represent heading-frames at particularpoints in time. The patterned circles at the centers of each dottedrectangles represent the robotic device's CoM with respect to itsheading-frame. The filled-in circles represent the locations of thestance leg, and the dotted lines connecting the filled-in circles to thepatterned circles represent the relative displacement between the CoMand the location of the stance leg in the x- and y-directions. The whitecircles represent the swing leg's touchdown location, and the dottedlines connected the white circles to the patterned circles represent therelative displacement between the CoM and the next touch-down locationfor the swing leg in the x- and y-directions of the associatedheading-frame. Note that the touchdown and lift-off locations aremeasured against different heading-frames. The dashed lines with thearrow running through the robotic device's CoM in each heading-framerepresents the robotic device's overall motion (e.g., the roboticdevice's CoM trajectory).

FIG. 7A illustrates a top-down view of an example straight-line walkingTDLO scenario. Moment 700 illustrates the position of the roboticdevice's CoM and the stance leg's touchdown position 712 at the momentwhen the stance leg makes initial contact with the ground. The touchdownposition 712 may be measured using a coordinate system defined by theheading-frame 710. The heading-frame 710 may be configured to move inthe positive x-direction, similarly to the constant-speed walkingexample described above.

After the robotic device's stance leg is planted on the ground attouchdown position 712, the robotic device's CoM moves forward (e.g., inthe positive x-direction) over that planted stance leg until moment 720.Moment 720 illustrates the position of the robotic device's CoM and thestance leg's lift-off position 732 at the moment when the stance legbegins to lift off the ground. The lift-off position 732 may be measuredusing a coordinate system defined by the heading-frame 730. In thisexample, the coordinate systems between heading-frame 710 andheading-frame 730 have the same orientation (although they aretranslated by an amount proportional to the change in position of therobotic device's CoM). Note that, with respect to absolute position(e.g., “world” position), the touchdown position 712 and the lift-offposition 732 may be the same location on the ground.

In the straight-line walking example, the touchdown position 712 and thelift-off position 732 measured against their respective heading-framesmay have the same amount of y-directional displacement, but differingx-directional displacement. Thus, when these foot measurements areapplied to TDLO, the swing leg's touchdown position 734 may be locatedsuch that stepping at that location causes the robotic device's CoM tomove in the positive x-direction, while the relative y-position of therobotic device's CoM remains constant.

FIG. 7B illustrates a top-down view of an example TDLO-based walkingscenario with turning. Moment 750 illustrates a heading-frame 760 thathas a similar orientation to the heading-frame 710 at moment 700.Additionally, the touchdown position 762 measured with respect to theheading-frame 760 may be similar to the touchdown position 712 measuredwith respect to heading-frame 710.

However, unlike the example in FIG. 7A, the heading-frame in FIG. 7B isconfigured to rotate in a counterclockwise direction (from the top-downperspective) at a constant rate. Therefore, at moment 770, theheading-frame 780 is rotated compared to the heading-frame 760. As aresult, the stance leg's lift-off position 782 measured using thecoordinate system defined by heading-frame 780 is different than thelift-off position 732. When the touchdown position 762 and the lift-offposition 782 (measured relative to the rotated coordinate system definedby the heading-frame 780) are applied to TDLO control law, the swingleg's touchdown position 784 may be located such that stepping at thatlocation causes the robotic devices' CoM to move in both a positivex-direction and a positive y-direction (e.g., forward-motion turning).Note that the absolute position of the swing leg touchdown position 784may be different from the absolute position of the swing leg touchdownposition 734. Thus, as illustrates in FIG. 7B, rotating theheading-frame over time may induce TDLO control law to cause the robotto turn.

As a general matter, moving, rotating, translating, or otherwisemodifying the heading-frame (e.g., the frame of reference from which theTDLO measurements are recorded) may cause the robot to correct,compensate, or otherwise modify its future stepping locations. In someimplementations, the heading-frame may be translated or rotated in atime-synchronized manner in order to carry out a choreographed set ofmovements using TDLO control law. Any combination of rotations and/ortranslations may be applied to the heading-frame for any number of steps(or for any length of time) without departing from the scope of thepresent application.

One way to modify TDLO control law to implement the heading-framemodifications described above is to utilize a forcing function. Anexample forcing function may be a one- or two-dimensional offset (e.g.,in the x-direction and/or the y-direction) that effectively alters thelocation around which TDLO-based stepping stabilizes.

The following description provides some example mathematicalrelationships that may be used to extend TDLO control law to includevelocity control and/or yaw control. The mathematical relationships andequations described below are provided for explanatory purposes andillustrate one example forcing function-based implementation. Othermathematical relationships that depend upon different parameters and/ordifferent robotic models may also be used to implement theheading-frame-based TDLO control described herein.

During a turning operation, control quantities (such as foot positions)may be measured with respect to a yawing coordinate system. In someimplementations, one horizontal axis (e.g., the x-axis, or thelongitudinal axis) of this yawing coordinate system may nominally pointin the desired direction of travel for the robot. This coordinate systemmay be referred to herein as the “heading-frame.” In this paradigm, TDLOcontrol may be performed in two horizontal axes simultaneously (e.g.,the x-axis and y-axis, or the longitudinal axis and lateral axis). Thetouchdown position of the stance leg {x_(TD,k)y_(TD,k)} may be measuredwith respect to the heading-frame at the moment of touchdown, while thelift-off position of the stance leg {x_(LO,k)y_(LO,k)} may be measuredwith respect to the heading-frame at the moment of lift-off. During asteady-state turn, the heading-frames at touchdown and lift-off may notbe aligned, but may instead have a rotational yaw-offset proportional tothe turning rate and the amount of time between touchdown and lift-off.During this steady-state turning, the touchdown positions expressed withrespect to the heading-frame at touchdown converge to the followingrelationships shown in equations [10] and [11]:

$\begin{matrix}{{\overset{\_}{x}}_{TD} = {\frac{{\overset{˙}{x}}_{k}}{\lambda\sinh\left( {\lambda T_{s}} \right)}\left( {{\cosh\left( {\lambda T_{s}} \right)} - {\cos\theta_{z}}} \right)}} & \lbrack 10\rbrack\end{matrix}$ $\begin{matrix}{{\overset{¯}{y}}_{TD} = {\frac{{\overset{˙}{x}}_{k}}{\lambda\sinh\left( {\lambda T_{s}} \right)}\sin\theta_{z}}} & \lbrack 11\rbrack\end{matrix}$

Where T_(s) is the stance period duration (in seconds), θ_(z) is thetotal yaw change (in radians) occurring over T_(s), and {dot over(x)}_(k) is the forward speed (in m/s) at the moment of touchdown. Whenperforming TDLO with respect to a yawing heading-frame, the TDLO controlof equation [1] may be written as equations [12] and [13]:

$\begin{matrix}{\begin{bmatrix}x_{{TD},{k + 1}} \\y_{{TD},{k + 1}}\end{bmatrix} = {{J_{2x2}\left\lfloor {{{R_{Z}^{T}\left( \theta_{z} \right)}\begin{bmatrix}x_{{TD},k} \\y_{{TD},k}\end{bmatrix}} - \begin{bmatrix}x_{{LO},k} \\y_{{LO},k}\end{bmatrix}} \right\rfloor} - {B_{2x2}\left\lfloor {{{R_{Z}^{T}\left( \theta_{x} \right)}\begin{bmatrix}x_{{TD},k} \\y_{{TD},k}\end{bmatrix}} + \begin{bmatrix}x_{{LO},k} \\y_{{LO},k}\end{bmatrix}} \right\rfloor} + \begin{bmatrix}\Delta_{x} \\\Delta_{y}\end{bmatrix}}} & \lbrack 12\rbrack\end{matrix}$ $\begin{matrix}{{R_{Z}^{T}\left( \theta_{z} \right)} = \begin{bmatrix}{c\theta_{z}} & {{- s}\theta_{z}} \\{s\theta_{z}} & {c\theta_{z}}\end{bmatrix}} & \lbrack 13\rbrack\end{matrix}$

Here J_(2×2) and B_(2×2) are 2×2 matrices of TDLO gains, andcθ_(z)=cos(θ_(z)) and sθ_(z)=sin(θ_(z)). Given the steady-statetouchdown positions specified above, the TDLO forcing terms [Δ_(x),Δ_(y)] can be derived, as shown in equation [14]:

$\begin{matrix}{{\left\lfloor \begin{matrix}{\Delta x} \\{\Delta y}\end{matrix} \right\rfloor = \left\lbrack {\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix} - {\left( {K_{2 \times 2} - B_{2 \times 2}} \right){R_{Z}^{T}\left( \theta_{z} \right)}} + {\left( {K_{2 \times 2} + B_{2 \times 2}} \right)\ \begin{bmatrix}{- 1} & 0 \\0 & 1\end{bmatrix}}} \right\rbrack}\text{ }\begin{bmatrix}{\overset{\_}{x}}_{TD} \\{\overset{¯}{y}}_{TD}\end{bmatrix}} & \lbrack 14\rbrack\end{matrix}$

Forcing TDLO with these values [Δ_(x),Δ_(y)] (i.e., incorporating theforcing terms into TDLO control law) may cause TDLO to coordinate thehorizontal center-of-mass motion with the current heading direction. Inthis manner, TDLO control law may be extended to control the magnitudeand direction of the robotic device's CoM velocity. A computing device,human interface device, and/or another planning or control system mayspecify these forcing values—as either constant values or time-varyingvalues—in order to cause the robot to turn in a desired manner. Theresulting TDLO control law including the forcing function term may bewritten as shown below in equation [15]:

x _(TD,k+1) =j _(x)(x _(TD,k) −x _(LO,k))−b _(x)(x _(TD,k) +x_(LO,k))+Δ_(k[15])  [15]

The forcing function term Δ_(k) may, in some instances, be different foreach foot step. A set of Δ_(k) values may be specified that causes therobotic device to perform acyclic stepping motions. For example, theforcing function may specify an offset for a few stance periods k thatcauses the robotic device to turn during those stance periods k, butthen continue walking in a straight line thereafter. As another example,the forcing function may contain a set of values that cause the roboticdevice to step in accordance with a pre-determined stepping pattern(e.g., performing the “cha-cha” dance). Further, the forcing functionterm Δ_(k) may not only vary from step-to-step (e.g., for differentvalues of k), but may also vary during a given stance period k.Regardless of the particular implementation, the forcing function termOk may be a constant value, a set of values, and/or may vary betweenstance periods or over time.

Note that, in some implementations, the value of the forcing functionmay be generated based on a joystick input and/or other peripheralinput. For example, if a user pushes a joystick in a forward direction,the joystick or a computing device may generate a forcing function valueproportional to the extent to which the joystick was pushed forward.This input value may be provided to a robotic device's control system,which may modify the forcing function term in the control system's TDLOcontrol law. In this manner, a user may control the robotic device usingan input device.

The TDLO control principles described above may be implemented as a partof a larger control system and/or planning system for a robotic device.For example, a robotic device may have a perception system configured toobserve the environment and detect possible obstacles or objectsproximate to the robotic device. The robotic device's control systemand/or planning system-which may implement TDLO control as describedherein—may be configured to avoid obstacles and/or objects within itsplanned path or trajectory (e.g., objects that the robotic device maypossibly collide with). In some implementations, the planning system maybe configured to modify the forcing function in order to alter therobotic device's direction of motion (and resulting trajectory) to avoidcolliding with that object.

FIG. 8 illustrates an example computer-readable medium configuredaccording to at least some implementation described herein. In exampleembodiments, an example system can include one or more processors, oneor more forms of memory, one or more input devices/interfaces, one ormore output devices/interfaces, and machine readable instructions thatwhen executed by the one or more processors cause the system to carryout the various operations, tasks, capabilities, etc., described above.

As noted above, the disclosed procedures can be implemented by computerprogram instructions encoded on a computer-readable storage medium in amachine-readable format, or on other media or articles of manufacture.FIG. 8 is a schematic illustrating a conceptual partial view of acomputer program product that includes a computer program for executinga computer process on a computing device, arranged according to at leastsome implementations disclosed herein.

In one embodiment, the example computer program product 800 is providedusing a signal bearing medium 802. The signal bearing medium 802 mayinclude one or more program instructions 804 that, when executed by oneor more processors may provide operations or portions of the operationsdescribed above with respect to FIGS. 1-5 , FIGS. 6A, 6B, and 6C, andFIG. 7 . In some examples, the signal bearing medium 802 can be acomputer-readable medium 806, such as, but not limited to, a hard diskdrive, a Compact Disc (CD), a Digital Video Disk (DVD), read-only memory(ROM), random-access memory (RAM), flash memory, a digital tape, memory,etc. In some implementations, the signal bearing medium 802 can be acomputer recordable medium 808, such as, but not limited to, memory,read/write (R/W) CDs, RAY DVDs, etc. In some implementations, the signalbearing medium 802 can be a communication medium 810 (e.g., a fiberoptic cable, a waveguide, a wired communications link, etc.). Thus, forexample, the signal bearing medium 802 can be conveyed by a wirelessform of the communications medium 810.

The one or more program instructions 804 can be, for example, computerexecutable and/or logic implemented instructions. In some examples, acomputing device is configured to provide various operations or actionsin response to the program instructions 804 conveyed to the computingdevice by the computer readable medium 806, the computer recordablemedium 808, and/or the communications medium 810. In other examples, thecomputing device can be an external device in communication with adevice coupled to the robotic device.

The computer readable medium 806 can also be distributed among multipledata storage elements, which could be remotely located from each other.The computing device that executes some or all of the storedinstructions could be an external computer, or a mobile computingplatform, such as a smartphone, tablet device, personal computer, or awearable device, among others. Alternatively, the computing device thatexecutes some or all of the stored instructions could be a remotelylocated computer system, such as a server. For example, the computerprogram product 800 can implement operations discussed in thedescription of FIG. 5 .

The arrangements described herein are for purposes of example only. Assuch, those skilled in the art will appreciate that other arrangementsand other elements (e.g., machines, interfaces, operations, tasks,orders, and groupings of operations, etc.) can be used instead, and someelements may be omitted altogether according to the desired results.Further, many of the elements that are described are functional entitiesthat may be implemented as discrete or distributed components or inconjunction with other components, in any suitable combination andlocation, or other structural elements described as independentstructures may be combined.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopebeing indicated by the following claims, along with the full scope ofequivalents to which such claims are entitled. It is also to beunderstood that the terminology used herein is for the purpose ofdescribing particular embodiments only, and is not intended to belimiting.

1. A method comprising: controlling, by data processing hardware of alegged robot, a first leg of the legged robot to move to a firsttouchdown location on a surface; obtaining, by the data processinghardware, a first measurement indicating a distance between a center ofmass of the legged robot and the first leg when the first leg touchesdown at the first touchdown location; controlling, by the dataprocessing hardware, the first leg to rise from the surface; obtaining,by the data processing hardware, a second measurement indicating adistance between the center of mass of the legged robot and the firstleg when the first leg is raised from the surface; determining, by thedata processing hardware, a second touchdown location on the surface fora second leg of the legged robot based on the first measurement and thesecond measurement; and controlling, by the data processing hardware,the second leg of the legged robot to move to the second touchdownlocation on the surface.
 2. The method of claim 1, further comprising:determining, by the data processing hardware, an estimated velocity ofthe legged robot based on a difference between the first measurement andthe second measurement; determining, by the data processing hardware, anestimated acceleration of the legged robot based on a sum of the firstmeasurement and the second measurement; and determining, by the dataprocessing hardware, the second touchdown location based on theestimated velocity and the estimated acceleration.
 3. The method ofclaim 2, wherein determining the second touchdown location based on theestimated velocity and the estimated acceleration comprises determining,by the data processing hardware, a linear combination of the estimatedvelocity and the estimated acceleration.
 4. The method of claim 2,further comprising determining, by the data processing hardware, anexpected path of the legged robot based on the estimated velocity andthe estimated acceleration.
 5. The method of claim 4, furthercomprising: detecting, by the data processing hardware, an expectedcollision of the legged robot with an object based on the expected path;and adjusting, by the data processing hardware, the second touchdownlocation to avoid the expected collision.
 6. The method of claim 1,wherein a first height of the surface at the first touchdown location isdifferent from a second height of the surface at the second touchdownlocation.
 7. The method of claim 1, further comprising controlling, bythe data processing hardware, a body of the legged robot to move overthe first leg when the first leg is touched down at the first touchdownlocation.
 8. The method of claim 1, further comprising: detecting, bythe data processing hardware, a slip of the first leg; and adjusting, bythe data processing hardware, the second touchdown location to accountfor the slip.
 9. The method of claim 1, further comprising determining,by the data processing hardware, the second touchdown location based ona damping ratio indicating a rate to suppress a perturbation arisingfrom an external force.
 10. The method of claim 1, further comprising:receiving, by the data processing hardware, a command indicating adesired change of velocity for the legged robot; and determining, by thedata processing hardware, the second touchdown location based on thedesired change of velocity.
 11. A legged robot comprising: a body; aplurality of legs coupled to the body, the plurality of legs comprisinga first leg and a second leg; data processing hardware; and memoryhardware in communication with the data processing hardware, the memoryhardware storing instructions that when executed on the data processinghardware cause the data processing hardware to perform operationscomprising: controlling the first leg to move to a first touchdownlocation on a surface; obtaining a first measurement indicating adistance between a center of mass of the legged robot and the first legwhen the first leg touches down at the first touchdown location;controlling the first leg to rise from the surface; obtaining a secondmeasurement indicating a distance between the center of mass of thelegged robot and the first leg when the first leg is raised from thesurface; determining a second touchdown location on the surface for asecond leg of the legged robot based on the first measurement and thesecond measurement; and controlling the second leg of the legged robotto move to the second touchdown location on the surface.
 12. The leggedrobot of claim 11, wherein the operations further comprise: determiningan estimated velocity of the legged robot based on a difference betweenthe first measurement and the second measurement; determining anestimated acceleration of the legged robot based on a sum of the firstmeasurement and the second measurement; and determining the secondtouchdown location based on the estimated velocity and the estimatedacceleration.
 13. The legged robot of claim 12, wherein determining thesecond touchdown location based on the estimated velocity and theestimated acceleration comprises determining a linear combination of theestimated velocity and the estimated acceleration.
 14. The legged robotof claim 12, wherein the operations further comprise determining anexpected path of the legged robot based on the estimated velocity andthe estimated acceleration.
 15. The legged robot of claim 14, whereinthe operations further comprise: detecting an expected collision of thelegged robot with an object based on the expected path; and adjustingthe second touchdown location to avoid the expected collision.
 16. Thelegged robot of claim 11, wherein a first height of the surface at thefirst touchdown location is different from a second height of thesurface at the second touchdown location.
 17. The legged robot of claim11, wherein the operations further comprise controlling the body to moveover the first leg when the first leg is touched down at the firsttouchdown location.
 18. The legged robot of claim 11, wherein theoperations further comprise: detecting a slip of the first leg; andadjusting the second touchdown location to account for the slip.
 19. Thelegged robot of claim 11, wherein the operations further comprisedetermining the second touchdown location based on a damping ratioindicating a rate to suppress a perturbation arising from an externalforce.
 20. The legged robot of claim 11, wherein the operations furthercomprise: receiving a command indicating a desired change of velocityfor the legged robot; and determining the second touchdown locationbased on the desired change of velocity.